0132497468-Ch12_ISM
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An Introduction to Shear Strength
Chapter 12
CHAPTER 12 AN INTRODUCTION TO SHEAR STRENGTH OF SOILS AND ROCK
12-1. A granular material is observed being dumped from a conveyor belt. It forms a conical pile with about the same slope angle, 1.8 horizontal to 1 vertical. What is the angle of internal friction of this material? SOLUTION: tan
y x
1 o tan1 29.0 1.8
12-4. A direct shear test was conducted on a fairly dense sample of Franklin Falls sand from New Hampshire. The initial void ratio was 0.668. The shear box was 76 mm square, and initially the height of the specimen was 11 mm. The tabulated data were collected during shear. Compute the data needed and plot the usual curves for this type of test. SOLUTION: Assuming c = 0, the friction angle can be calculated from plot 3 as: 306.44 o '=tan1 38.2 389.54
continued on next page.
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-4 data table.
Shear Stress (kPa)
350 300 250 200 150 100 50 0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
3.5
4.0
Thickness Change (mm
Horz. Displacement (mm) 0.10 0.08 0.06 0.04 0.02 0.00 -0.02 -0.04 -0.06 0.0
0.5
1.0
1.5
2.0
2.5
3.0
Horz. Displacement (mm)
Shear Stress (kPa)
400
300
200
100
0 0
100
200
300
400
500
600
Normal Stress (kPa)
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-5. A conventional triaxial compression test was conducted on a sample of dense sand from Ft. Peck Dam, Montana. The initial area of the test specimen was 10 cm2 and its initial height was 70 mm. Initial void ratio was 0.605. The following data were observed during shear. First, calculate the average area of the specimen, assuming it is a right circular cylinder at all times during the test. Then make the calculations necessary to plot the axial stress versus axial strain and volumetric-strain-versus-axial-strain curves for this test. Assuming c’ = 0, what is ’?
SOLUTION:
Average H 66.887 mm, 1f 983 kPa, Eq. (11.13)
Volume 70 cm3
3f 206.8 kPa
sin
1f 3f 1f 3f
983 206.8 o sin1 40.7 983 206.8
continued on next page.
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-5 continued.
Axial Stress (kPa
1200.00 1000.00 800.00 600.00 400.00 200.00 0.00 0.0
2.0
4.0
6.0
8.0 10.0 12.0 14.0 16.0
Axial Strain (%)
Volumetric Strain (%)
10.000 5.000 0.000 0.0
2.0
4.0
6.0
8.0 10.0 12.0 14.0 16.0
-5.000 -10.000 -15.000
Axial Strain (%) 800
M-C failure envelope
600
Shear stress (kPa
400
200
0 206.8
983
-200
-400
-600
-800 0
200
400
600
800
1000
1200
1400
1600
Normal stress (kPa)
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-6. The results of two CD triaxial tests at different confining pressures on a medium dense, cohesionless sand are summarized in the table below. The void ratios of both specimens were approximately the same at the start of the test. Plot on one set of axes the principal stress difference versus axial strain and volumetric strain [Eq. (12.4)] versus axial strain for both tests. Estimate the initial tangent modulus of deformation, the “50%” secant modulus, and the strain at failure for each of these tests.
SOLUTION: Test 1 325 kPa 19,006 kPa 0.0171 Evaluate Esec at 50% of max For this test Esec Et 19, 600 kPa
Et
Test 2 1500 kPa 375,000 kPa 0.004 Evaluate Esec at 50% of max Et
9140 4570 152,333 kPa =4570 kPa; Esec 2 0.03 Re fer to plots on next page. 50% =
continued on next page
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An Introduction to Shear Strength
Chapter 12
12-6 continued.
Deviator Stress (kPa
10000 9000
Test 1 at 100 kPa
8000
Test 2 at 3000 kPa
7000 6000 5000 4000 3000 2000 1000 0 0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
14.0
16.0
Axial Strain (%)
Volumetric Strain (%)
8.0
Test 1 at 100 kPa 6.0
Test 2 at 3000 kPa
4.0 2.0 0.0 0.0
2.0
4.0
6.0
8.0
10.0
12.0
-2.0 -4.0 -6.0
Axial Strain (%)
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-7. For the two tests of Problem 12.6, determine the angle of internal friction of the sand at (a) peak compressive strength, (b) at ultimate compressive strength, and (c) at 5.5% axial strain. SOLUTION: sin
Eq. (11.13)
1f 3f 1f 3f
Test 1 (a) Peak:
1f 3f 441,
(b) Ultimate:
441 o peak sin1 43.5 541 100 308 o 1f 308 100 408, peak sin1 37.3 408 100
1f 441 100 541,
1f 3f 308,
(c) At 5.5% strain:
1f 3f 440,
1f 440 100 540,
440 o peak sin1 43.4 540 100
Test 2 (a) Peak:
1f 3f 9140,
(b) Ultimate:
1f 3f 9090,
(c) At 5.5% strain:
9140 o peak sin1 37.1 12,140 3000 9090 o 1f 9090 3000 12,090, peak sin1 37.0 12,090 3000
1f 9140 3000 12,140,
1f 3f 6800,
1f 6800 3000 9800,
6800 peak sin1 32.1o 9800 3000
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-8. A sand is hydrostatically consolidated in a triaxial test apparatus to 450 kPa and then sheared with the drainage valves open. At failure, (1 – 3) is 1121 kPa. Determine the major and minor principal stresses at failure and the angle of shearing resistance. Plot the Mohr diagram. (This problem should be followed by the next one.) SOLUTION: Eq. (11.13)
sin
Peak: 3f 450,
1f 3f 1f 3f 1f 3f 1121,
1f 1121 450 1571
1121 o peak sin1 33.69 1571 450
800
600
Shear stress (kPa
400
200 450
0
1571
-200 0 -400
200
400
600
800
1000
1200
1400
1600
Normal stress (kPa)
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-9. The same sand as in Problem 12.8 is tested in a direct shear apparatus under a normal pressure of 390 kPa. The specimen fails when a shear stress of 260 kPa is reached. Determine the major and minor principal stresses at failure and the angle of shearing resistance. Plot the Mohr diagram. SOLUTION: Plot (390, 260) and draw failure envelope assuming c = 0. Extend perpendicular line to x axis to locate circle center. Calculate radius and center using geometry. 260 ' tan1 33.69 390 390 cos 33.69 a 468.72 a 468.72 C 563.33 cos 33.69 C R R 312.48 tan 33.69 468.72 3 C R 563.33 312.48 250.8 kPa 1 C R 563.33 312.48 875.81 kPa
800
600
400
Shear stress (kPa
(390, 260) 200 250.8
0
450
563.33
875.81
1571
-200 0 -400
200
400
600
800 1000 1200 Normal stress (kPa)
1400
1600
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-10. Indicate the orientations of the major principal stress, the minor principal stress, and the failure plane of the tests in Problems 12.8 and 12.9. SOLUTION: Plot (390, 260) and draw failure envelope assuming c = 0. Extend perpendicular line to x axis to locate circle center. Calculate radius and center using geometry. (a) Direct Shear Test Pole = (736.66, 260) 260 p3 tan1 28.15 measured ccw from horizontal 736.66 250.8 p1 180 90 28.15 61.85 measured cw from horizontal The failure plane is horizontal. (b) CD Triaxial Test Pole = (450, 0) 3 acts on the vertical plane, 1 acts on the horizontal plane 45
' 33.69 45 61.85 measured ccw from the horizontal 2 2
800
600
400 Pole=(736.66, 260)
Shear stress (kPa
(390, 260) 200
875.81
250.8
0
1571
Pole=(450, 0)
-200 0 -400
200
400
600
800
1000
1200
1400
1600
Normal stress (kPa)
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-11. A granular soil is tested in direct shear under a normal stress of 350 kPa. The size of the specimen is 7.62 cm in diameter. If the soil to be tested is a dense sand with an angle of internal friction of 38°, determine the size of the force transducer required to measure the shear force with a factor of safety of 2 (that is, the capacity of the transducer should be twice that required to shear the sand). SOLUTION: tan ' (350) tan 38 273.4 Fshear 2 546.9 kPa
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-12. The stresses induced by a surface load on a loose horizontal sand layer were found to be v = 5.13 kPa, v = 1.47 kPa, h = 3.2 kPa, h = -1.47 kPa. By means of Mohr circles, determine if such a state of stress is safe. Use Eq. (11.11) for the definition of factor of safety. SOLUTION: Eq. (11.13)
sin
1f 3f 1f 3f
From the Mohr circle: C 4.175, R 1.753,
1 5.93,
3 2.42,
max 1.753
5.93 2.42 o (on the failure plane, but not at failure) sin1 24.8 5.93 2.42 (C x) sin(90 ) f ; cos(90 ) ; f C x R R f (1.753) sin(90 24.8) 1.59 f 4.175 (1.753) cos(90 24.8) 4.175 0.735 3.44 For loose sand, assume ' = 30 ff (3.44) tan 30 1.99 (Eq. 11.11) FS
ff 1.99 1.25 f 1.59
3.0 2.5 3.44, 1.99
2.0 1.5
3.44, 1.59
5.13, 1.47
Shear stress (kPa
1.0 0.5 0.0
2.42
5.93
-0.5 -1.0 -1.5
3.22, -1.47
-2.0 -2.5 -3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Normal stress (kPa)
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-13. If the same stress conditions as in Problem 12.12 act on a very dense gravelly sand, is such a state safe against failure? SOLUTION: Eq. (11.13)
sin
1f 3f 1f 3f
From the Mohr circle: C 4.175, R 1.753,
1 5.93,
3 2.42,
max 1.753
5.93 2.42 o (on the failure plane, but not at failure) sin1 24.8 5.93 2.42 (C x) sin(90 ) f ; cos(90 ) ; f C x R R f (1.753) sin(90 24.8) 1.59 f 4.175 (1.753) cos(90 24.8) 4.175 0.735 3.44 For very dense gravelly loose sand, assume ' = 38 ff (3.44) tan 38 2.69 (Eq. 11.11) FS
ff 2.69 1.69 f 1.59
3.0
3.44, 2.69
2.5 2.0 1.5
3.44, 1.59
5.13, 1.47
Shear stress (kPa
1.0 0.5 0.0
2.42
5.93
-0.5 -1.0 -1.5
3.22, -1.47
-2.0 -2.5 -3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Normal stress (kPa)
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-14. The effective normal stresses acting on the horizontal and vertical planes in a silty gravel soil are 1.77 MPa and 2.95 MPa, respectively. The shear stress on these planes is 0.59 MPa. For these conditions, what are the magnitude and direction of the principal stresses? Is this a state of failure? SOLUTION: Eq. (11.13)
sin
1f 3f 1f 3f
From the Mohr circle: C 2.36, R 0.834,
1 3.19,
3 1.52,
max 0.834
3.19 1.52 o sin1 (on the failure plane, but not at failure) 20.77 3.19 1.52 The given state of stress is not in a state of failure, because ' for this material > 20.8. 1 3.19 MPa oriented 67.9° ccw from horizontal 3 1.52 MPa
oriented 22.1° cw from horizontal
2.0
1.5
Shear stress (MPa
1.0
0.5 1.53
0.0
3.19
-0.5 Pole (2.95, 0.59) -1.0
-1.5
-2.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Normal stress (MPa)
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An Introduction to Shear Strength
Chapter 12
12-15. A specimen of dense sand tested in a triaxial CD test failed along a well-defined failure plane at an angle of 62° with the horizontal. Find the effective confining pressure of the test if the principal stress difference at failure was 115 kPa. SOLUTION: ' 62 2 1 3 f 115 kPa 45
' 34
115 57.5 2 R 57.5 C 102.8 sin ' C sin 34 1 C R 102.8 57.5 160.3
R
3 C R 102.8 57.5 45.3 kPa
12-16. A dry loose sand is tested in a vacuum triaxial test in which the pore air pressure of the specimen is lowered below gage pressure to within about 95% of -1 atm. Estimate the principal stress difference and the major principal stress ratio at failure. SOLUTION: 1atm 14.7 psi (0.95)( 14.7) 13.96 psi confining pressure 3 = (atm pressure) - (vacuum pressure) 3 13.96 psi For loose sand, assume ' 30 (Eq. 11.16)
1 tan2 45 3 2
30 1 (13.96) tan2 45 41.88 psi 2 1f 3f 41.88 13.96 27.92 psi 1f 41.88 3.0 3f 13.96
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An Introduction to Shear Strength
Chapter 12
12-17. For the data shown in Fig. 12.5(a), what is (a) the principal stress difference and (b) the principal stress ratio at an axial strain of 12% for an effective confining pressure of 1.3 MPa?
SOLUTION: Given : 12% and c 3 1.3 MPa (a)
1 3.1 3
1 (3.1)(1.3) 4.0 MPa
(b) 1 3 4.0 1.3 2.7 MPa
12-18. For the conditions given in Problem 12.17, plot the Mohr circle. SOLUTION: C 2.65, R 1.35 2.0
1.5
Shear stress (MPa
1.0
0.5 1.30
0.0
4.00
-0.5
-1.0
-1.5
-2.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Normal stress (MPa)
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-19. Do Problems 12.17 and 12.18 for the data shown in Fig. 12.6(a). Use c = 1.0 MPa.
SOLUTION: Given : 12% and c 3 1.0 MPa (a)
1 4.1 3
1 (4.1)(1.0) 4.1MPa
(b) 1 3 4.1 1.0 3.1 MPa Mohr Circle: C 2.55, R 1.55 2.0
1.5
Shear stress (MPa
1.0
0.5 1.00
0.0
4.10
-0.5
-1.0
-1.5
-2.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Normal stress (MPa)
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-22. A drained triaxial test is performed on a sand with ’3c=’3f = 450 kPa. At failure, max = 594 kPa. Find ’1f, (1 – 3)f,, and ’. SOLUTION: radius max 594,
'3f 450
center 450 594 1044 '1f 450 2(594) 1638 kPa
1 3 f 1638 450 1188 Eq. (11.13)
sin
kPa
1f 3f 1f 3f
1188 sin1 34.68o 1638 450
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-23. Assume the sand of Problem 12.22 is Sacramento River sand at a void ratio of 0.6. If the initial volume of the specimen was 62 cm3, what change in volume would you expect during shear? SOLUTION: radius max 594,
'3f 450
center 450 594 1044 '1f 450 2(594) 1638 kPa
1 3 f 1638 450 1188 '1f 1.638 MPa,
kPa
'3f 0.45 MPa
1f 1.638 3.64 3f 0.45 From Fig. 12.6a, estimate 1.5% U sin g Fig. 12.6b, for '3f 0.45 MPa and 1.5% vol
V Vo
Vol. strain, vol 1.0%
V (62)(0.01) 0.62 cm3 (dilation)
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-24. A silty sand is tested consolidated-drained in a triaxial cell where both principal stresses at the start of the test were 625 kPa. If the total axial stress at failure is 2.04 MPa while the horizontal pressure remains constant, compute the angle of shearing resistance and the theoretical orientation of the failure plane with respect to the horizontal. SOLUTION: radius max 594,
'3f 450
center 450 594 1044 '1f 450 2(594) 1638 kPa
1 3 f 1638 450 1188 Eq. (11.13)
sin
kPa
1f 3f 1f 3f
1188 sin1 34.68o 1638 450
12-25. A specimen of sand failed when (1 – 3) was 750 kPa. If the hydrostatic consolidation stress was 250 kPa, compute the angle of shearing resistance of the sand. What else can you say about the sand? SOLUTION: '3f 250,
1 3 f 750
center 450 594 1044 '1f 750 250 1000 Eq. (11.13)
sin
1f 3f 1f 3f
750 o sin1 36.9 1000 250
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-26. A specimen of sand at the field density is known to have a (1/3)max of 3.8. If such a specimen is hydrostatically consolidated to 1180 kPa in a triaxial test apparatus, at what effective confining pressure will the specimen fail if the vertical stress is held constant? (This is a lateral extension test.) SOLUTION: 1f 3.8, 3f
'3f 1180
Eq. (11.13)
sin
1f (1180)(3.8) 4484
1f 3f 1f 3f
4484 1180 sin1 35.68 4484 1180 The effective confining pressure 4484 1180 3304 kPa
12-27. Two CD triaxial tests are conducted on identical specimens of the same sand. Both specimens are initially consolidated hydrostatically to 50 kPa; then each specimen is loaded as shown. Specimen A failed when the applied 1 was 180 kPa. Make the necessary calculations to (a) plot the Mohr circles at failure for both tests, and (b) determine ’ for the sand.
SOLUTION: '1f 230 kPa, Eq. (11.13)
'3f 80 kPa
sin
1f 3f 1f 3f
230 80 o sin1 28.9 230 80
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-28. Plot a graph of ’1/’3 versus ’. SOLUTION:
Princ. stress ratio
'1 ' tan2 45 '3 2 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 20
25
30
35
40
Friction angle (deg)
12-29. Estimate the shear strength parameters of a fine (beach) sand (SP). Estimate the minimum and maximum void ratios. SOLUTION: ’ depends on relative density among a number of other items as described in Section 12.5. (c’ = 0.)
Friction angle, ’, for Dr from 50% to 75% ranges from about 32o to 36o, respectively. (see Fig. 12.15) Void ratio, e, for Dr from 50% to 75% ranges from about 0.85 to 0.76, respectively.
12-30. A subrounded to subangular sand has a D10 of about 0.1 mm and a uniformity coefficient of 3. The angle of shearing resistance measured in the direct shear test was 47°. Is this reasonable? Why or why not? SOLUTION:
This is not a reasonable value of ’. From Fig. 12.15, this value of ’ would be more applicable to a very dense gravel.
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-31. Estimate the ’ values for (a) a well-graded sandy gravel (GW) at a density of 1.9 Mg/m3; (b) a poorly graded silty sand with a field density of 1.70 Mg/m3; (c) an SW material at 100% relative density; and (d) a poorly graded gravel with an in situ void ratio of 0.5. SOLUTION:
Estimate ranges of ’ values using Fig. 12.15. (a) ’ = 40 to 45 deg (Fig. 12.15) (b) ’ = 34 to 38 deg (Fig. 12.15) (c) ’ = 41 deg (Fig. 12.14) (d) ’ = 41 deg (Fig. 12.13), ’ = 28 deg (Fig. 12.14)
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An Introduction to Shear Strength
Chapter 12
12-32. The results of a series of CD triaxial tests on a medium dense, cohesionless sand are summarized in the table below. The void ratios for all the test specimens were approximately the same at the start of the test. Plot the strength circles and draw the Mohr failure envelope for this series of tests. What angle of internal friction should be used in solving stability problems in which the range of normal stresses is (a) 0–500 kPa; (b) 1000–1500 kPa; (c) 3–6 MPa; and (d) 0–6 MPa?
SOLUTION:
3
1-3
1
'
(kPa)
(kPa)
(kPa)
(deg)
Test No. 1
120
576
696
44.90
2
480
2240
2720
44.43
3
1196
4896
6092
42.21
4
2256
8460
10716
40.71
5
3588
12240
15828
39.08
6
3568
15228
18796
42.92
Eq. (11.13)
sin
1f 3f 1f 3f
The failure envelope is curved over the large range of normal stress used in the tests. Eq. 11.13 provides the friction angle for each individual test; however, the tests should be considered on the aggregate over the normal stress ranges provided in the problem statement. (a) ' 44 (b) ' 43 (c) ' 41 (d) ' 42
Mohr circles shown on the next page.
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An Introduction to Shear Strength
Chapter 12
12-32 continued.
5000
Shear stress (kPa)
3000
1000
-1000
-3000
-5000 0
2000
4000
6000
8000
10000
Normal stress (kPa) 10000
Shear stress (kPa)
5000
0
-5000
-10000 0
5000
10000
15000
20000
Normal stress (kPa)
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An Introduction to Shear Strength
Chapter 12
12-33. Estimate the values of the coefficient of earth pressure at rest, Ko, for the four soils of Problem 12-31. SOLUTION: (Eq. 12.8) K o 1 sin '
(a) ’ = 40 to 45 deg (Fig. 12.15); Ko = 0.36 to 0.29 (b) ’ = 34 to 38 deg (Fig. 12.15); Ko = 0.0.44 to 0.38 (c) ’ = 41 deg (Fig. 12.14); Ko = 0.0.34 (d) ’ = 41 deg or ’ = 28 deg; Ko = 0.0.34 or Ko = 0.0.53
12-34. If the sands of Problem 12.33 had been preloaded, would your estimate of be any different? If so, would it be higher or lower? Why? SOLUTION:
Ko would be higher. Refer to Eq. 12.9, Section 12.7, for a discussion regarding the influence of preloading on Ko.
12-35. Estimate Ko for sands 1, 4, 5, 6, 8, and 10 in Table 12.1 for relative densities of 40% and 85%. SOLUTION: (Eq. 12.8) K o 1 sin '
Sand No.
(Loose, Dr = 40%) ' Ko
(Dense, Dr = 85%) ' Ko
(deg)
(deg)
1
28
0.531
35
0.426
4
33
0.455
37
0.398
5
36
0.412
40
0.357
6
34
0.441
42
0.331
8
35
0.426
46
0.281
10
38
0.384
47
0.269
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An Introduction to Shear Strength
Chapter 12
12-38. A CD axial compression triaxial test on a normally consolidated clay failed along a clearly defined failure plane of 54°. The cell pressure during the test was 220 kPa. Estimate ’, the maximum principal stress ratio and the principal stress difference at failure. SOLUTION: 54 45
' 2
' 18
'1 18 ' tan2 45 tan2 45 1.89 '3 2 2 '3 220, '1 (1.89)(220) 416.8
'1 '3 416.8 220 196.8
kPa
12-39. An unconfined compression test is performed on a dense silt. Previous drained triaxial tests on similar samples of the silt gave ’ = 32o. If the unconfined compressive strength was 420 kPa, estimate the height of capillary rise in this soil above the ground water table. (Hint: Find the effective confining pressure acting on the specimen. Draw elements similar to Fig. 12.40.) SOLUTION: '1f 230 kPa,
'3f 80 kPa
'1f '3f '1f '3f '1f '3f sin ' '1f '3f f ur uf ur uf sin ' f ur uf ur uf f 2ur 2uf sin ' f
Eq. (11.13)
sin
f 420,
' 32
(Eq. 12.16)
u f B
1 1 1 3 (1) (420) 140 3 3
420 2ur 2(140) sin 32 140 2ur sin 32 230.94 ur 217.9 kPa hc
ur 217.9 0.0222 m 22.2 mm w (1000)(9.81)
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
An Introduction to Shear Strength
Chapter 12
12-43. The results of unconfined compression tests on a sample of clay in both the undisturbed and remolded states are summarized below. Determine the compressive strength, the initial tangent modulus of deformation, and the secant modulus of deformation at 50% of the compressive strength for both the undisturbed and remolded specimens. Determine the sensitivity of the clay. What shear strength would you use?
SOLUTION: Solutions obtained from the stress-strain plot shown below. Undisturbed State compressive strength = 153 kPa Et
31 79.5 3100 kPa; E50 2732 kPa 0.01 0.028
Remolded State compressive strength = 48 kPa Et
24 7 700 kPa; E50 571 kPa 0.042 0.01
153 48 76.5 kPa; f (remolded) 24 kPa 2 2 (undisturbed) 76.5 Sensitivity f 3.2 24 f (remolded) f (undisturbed)
undisturbed shear strength, f 76.5 kPa 180
undisturbed state
Deviator Stress (kPa
160
remolded state
140 120 100 80 60 40 20 0 0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
Axial Strain (%)
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An Introduction to Shear Strength
Chapter 12
12-45. For the data shown in Fig. 8.5, estimate the unconfined compressive strength and the sensitivity of this soil. Typical values for the clay are LL = 88, PL = 43, and PI = 45. SOLUTION: w PL (Natural water content is needed to calculate LI) PI Figure 12.48 can be used to correlate S t to LI. (Eq. 2.40) LI
© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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